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1
Incorporation of Topology Optimization Capability in MSC/NASTRAN
GaoWen YE
Manager, Technical Department
MSC Japan, Ltd.
2-39, Akasaka 5-Chome, Minato-ku, Tokyo 107-0052, Japan
Keizo ISHII President & CEO
Quint Corporation
1-14-1 Fuchu-cho, Fuchu, Tokyo 183, Japan
ABSTRACT
In recent years, CAE based optimization applications have gained a wide acceptance in various structure or
component design. In MSC/NASTRAN, various design sensitivity and optimization capabilities, such as
sizing and shape optimization, have been introduced and continually enhanced. In order to provide all
MSC/NASTRAN users with a more complete spectrum of design optimization analysis capabilities,
including topology or layout optimization capability used in the very beginning of design process, the
topology optimization optimizer function of OPTISHAPE, a topology optimization program developed by
Quint Corporation, has been successfully incorporated into MSC/NASTRAN by the joint efforts of MSC
Japan, Ltd. and Quint Corporation. This paper describes the new topology optimization capability in
MSC/NASTRAN, and includes several application examples.
2
1 Introduction Design optimization is used to produce a design that possesses some optimal characteristics, such as
minimum weight, maximum first natural frequency, or minimum noise levels. Design optimization is
available in MSC/NASTRAN SOL 200, in which a structure can be optimized considering simultaneous
static, normal modes, buckling, transient response, frequency response, aeroelastic, and flutter analyses.
In SOL 200, both sizing parameters (the dimension of cross-section of beam elements like the height and
width, or the thickness of shell elements) and shape (grid coordinates related) parameters can be used as
design variables.
However, these optimization capabilities are basically only usable to improve the design of structures or
parts in the detailed design process. In the very beginning of the conceptual design process, another
category of optimization capability, so-called topology or layout optimization is necessary.
In order to provide all MSC/NASTRAN users with a more complete spectrum of design optimization
analysis capabilities, the topology optimization optimizer function of OPTISHAPE, the first commercial
code and the most famous topology optimization program developed by Quint Corporation, has been
successfully incorporated into MSC/NASTRAN, and will be called MSC/NASTRAN-OPTISHAPE. The
joint development of MSC/NASTRAN-OPTISHAPE by MSC Japan, Ltd. and Quint Corporation started
in the Summer of 1998 and is now commercially available through MSC Japan, Ltd. This paper will
describe MSC/NASTRAN-OPTISHAPE, and will include various application examples.
2 Topology Optimization Based on Homogenization Method
2.1 Basic Concept
MSC/NASTRAN-OPTISHAPE is based on a structural topology optimization approach, using the
homogenization method which was introduced by Bendsoe and Kikuchi [1] in 1988 as the theory of
optimal design of material distribution in the design domain. This theory has been extended and applied
to various kinds of problems such as static [2] and normal modes [3]. In topology optimization of elastic
structures based on this theory, the design domain is assumed to be composed of infinitely periodic
microstructures. In the case of two-dimensional problems, each micro-structure has a rectangular hole as
shown in Fig. 1. In the optimization process, the hole sides and angle of rotation, a , b and θ,
respectively, are taken as design variables, which are to be determined by minimizing/maximizing the
objective function subject to volume constraint and boundary conditions as mentioned below. Since
3
each element hole is allowed to possess a different size and angle of rotation, uniformly distributed
porous material in the initial stage will have a different size of element holes at the end of optimization
as shown in Fig. 1. Therefore, if the domain is viewed in a global sense, an obviously different resultant
topology can be obtained at the end of design process.
Figure 1 Concept of Topology Optimization of Using the Homogenization Method
2.2 The Homogenization Method
In the topology optimization approach based on finite element analysis, material properties of porous
material with various hole sizes are needed for both structural analysis and sensitivity analysis. In
MSC/NASTRAN-OPTISHAPE, the homogenized material constants of porous material is calculated by
the homogenization method. The homogenization method is promising in that this method gives
homogenized material constants of the composite material without any empirical assumptions, if the
material properties of all the constitutive materials are known. In this method it is usually assumed that
the composite material is locally formed by very small, periodical microstructures compared with the
overall macroscopic dimensions of the structure of interest. In such cases, the material properties are
periodical functions of microscopic variables when the period of the microstructures is very small
compared with the macroscopic variables. In the structural topology optimization using the
homogenization method, the periodical microstructure of porous material is usually called a unit cell. In
MSC/NASTRAN-OPTISHAPE, three kinds of unit cells are provided corresponding to both
two-dimensional problems and three-dimensional problems as shown in Fig. 2.
Design domain Ω
1
1
a
θ
b
y1
y2
x1
x2
4
2D-Shell ‘Composite’ Shell 3D-SOLID
Figure 2 Unit Cells Used in MSC/NASTRAN-OPTISHAPE
2.3 Optimization Problem Statements
In MSC/NASTRAN-OPTISHAPE, the optimal topology of a structure with the highest stiffness or the
highest/desired eigenvalues is calculated by changing the hole sizes as expressed by the following
optimization problem statements.
2.3.1 Static Problem
Minimize the mean compliance:
2.3.2 Eigenvalue Problem – Case 1
Maximize the mean eigenfrequencies:
2.3.3 Eigenvalue Problem – Case 2
Minimize the distance between the desired eigenfrequencies and the calculated ones:
( ) ( )
C
m
ii
m
iiii
VdtsMinimize ≤ΩΛ
−=Λ
∫
∑∑
Ω
==
ρ
λλλλ
..
11
20
1
20
20
C
HT
VdtsMinimize
dD
≤ΩΦ
Ω=Φ
∫∫
Ω
Ω
ρ
εε
..21
C
m
i i
im
ii
VdtsMaximize
ww
≤ΩΛ
=Λ
∫
∑∑
Ω
==
ρ
λ
..
11
5
3 Basic Design Notes
In the first phase of this development, the most commonly used features of OPTISHAPE, both static and
normal modes topology optimization analyses for either 2D or 3D problems, are introduced into
MSC/NASTRAN. In 2D problem analysis, the “composite” shell design capability of OPTISHAPE is
also implemented, in which only the 2 outer plies of 3-ply composite are going to be designed with the
base middle ply remaining the same. In order to develop an efficient and fully integrated solution
sequence for the topology optimization analysis, the optimizer function of OPTISHAPE is extracted and
rewritten as the topology optimization optimizer function modules of MSC/NASTRAN, and a new
solution sequence, TOPOPT, is developed by modifying the DMAP of both SOL 1 and SOL 3 and adding
some special new modules necessary for the purpose of topology optimization and efficiency. Figure 3
shows the conceptual flowchart of the TOPOPT, where TOPOPT is the solution sequence name, and
TOP1, TOP2, TOPDTI, TOPSDR, TOPMPT, TOPEID and TOPDEN are newly created function modules.
Either static or normal modes topology optimization can be performed by this sequence.
Figure 3 Conceptual Flowchart of TOPOPT
Yes
MSC/NASTRAN TOPOPT
Initial data pre-processing and datablocks preparation
TOPDTI, TOPEID, TOP1
Prepare special topology optimization datablocks & modify initial material table
TOPMPT, EMG, EMA, …,SSGi, SDRi, ELDFDR
Form stiffness & mass matrices and solve equations for disp., stress, ...
Converged / Max design cycles cyc Exit
TOPSDR, TOP2, TOPDEN
Sensitivity calculation & update the material table tabletable
6
4 Additional Data Input Description For the purpose of performing static or normal modes topology optimization, a few additional parameters
or data, such as constraint volume, move limit, design domain, etc., are necessary in addition to the
common MSC/NASTRAN static or normal modes analysis bulk data. In this version, these additional
data are provided by using MSC/NASTRAN DTI (Direct Table Input) entry as described below. This
entry can be inserted into MSC/NASTRAN bulk data file manually with some text editor like unix vi, or
within MSC/PATRAN by using MSC/NASTRAN-OPTISHAPE preference.
In most static topology optimization cases, only 4 (if without composite material design) or 6 (if with
composite material design) additional data lines are sufficient if one has the associated static analysis
data deck. Please also refer to application example 1 below to have a better understanding.
Table 1 DTI Input of TOPOL Data Block
DTI Format
$ 0 record, control scalar variables
1 2 3 4 5 6 7 8 9 10
DTI TOPOL 0 TBFLG
KANALY IOPT ITERO ITERX CVOL XCI OPTCOV OCYCLE
KOBJ MULTIE
$ 1 record, design elements by property or element list
DTI TOPOL 1 POEFLG
POEIDi POEIDj POEIDk … ENDREC
$ 2 record, shell/plate base layer thickness (k=2 if TBFLG=1)
DTI TOPOL k
TBi TBj TBk … ENDREC
$ 3 record, normal modes topology optimization inputs (k=2 if TBFLG=0, k=3 otherwise)
DTI TOPOL k
MODEi EIGRi WGTi MODEj EIGRj WGTj
MODEk EIGRk WGTk … … … ENDREC
7
Field Contents TBFLG = 1 if TBi are present.
= 0 if TBi are not present.
KANALY Kind of analysis.
= 1 : static topology optimization.
= 2 : eigenvalue topology optimization.
IOPT Kind of elements to be optimized.
= 2 : shell elements ( CTRIA3, CQUAD4 ).
= 3 : solid elements ( CTETRA, CPENTA, CHEXA ).
ITER0 Start cycle ( = 1 in this version)
ITERX Maximum allowable number of design cycles to be performed.
CVOL Constraint volume ( 0.01 < cvol < 0.99, default = 0.5 ) .
XCI Maximum move limit imposed ( 0.01 < xci < 0.5, default = 0.3).
OPTCOV Relative criterion to detect convergence (default = 0.001).
OCYCLE Output element volume density at every n-th cycle.
(default = -1, only output the element volume density at the last cycle.)
If OCYCLE>0, then the element volume density will be output at first cycle;
at every design cycle that is a multiple of OCYCLE; and the last design cycle.
KOBJ Kind of objective function for normal modes topology optimization.
= 1, maximize mean eigenfrequencies.
= 2, minimize distances between input eigenfrequencies and computed ones.
MULTIE Total number of eigenfrequencies to be considered.
POEFLG poeflg=0, specify design elements with element ID list.
poeflg=1, specify design elements with property ID list.
POEIDi Element or property list of design elements.
TBi Base layer thickness if TBFLG=1. If provided, the same number of TBi as POEIDi must be given.
MODEi Mode number to be considered.
EIGRi Desired frequencies (not necessary in case of KOBJ=1 ).
WGTi Weighting factors (default = 1.0).
5 MSC/PATRAN MSC/NASTRAN-OPTISHAPE Preference
As described above, in addition to the common MSC/NASTRAN data deck, only a few data lines should
be modified and added for the topology optimization. In order to do this with MSC/PATRAN, a special
8
MSC/PATRAN preference, MSC/NASTRAN-OPTISHAPE preference, is developed by adding some
topology optimization specific data pre and post processing part to the existing MSC/PATRAN
MSC/NASTRAN preference. The major functions of the preference are as follows:
1) Define MSC/NASTRAN-OPTISHAPE analysis parameters, specify design domain for optimization
and generate MSC/NASTRAN-OPTISHAPE specific data based on an original MSC/NASTRAN
analysis job and analysis bulk data deck.
2) Read existing OPTISHAPE analysis deck into database.
3) Read MSC/NASTRAN-OPTISHAPE specific results (element volume density) into database.
4) Postprocess MSC/NASTRAN-OPTISHAPE specific results (element volume density).
To efficiently define the design domain (i.e. to define the elements that are to be optimized), the
preference will use property set names to indicate design elements indirectly, instead of specifying
design elements directly. Figure 4 shows several main menus of the newly developed
MSC/NASTRAN-OPTISHAPE preference.
Figure 4 Several Main Menus of MSC/NASTRAN-OPTISHAPE Preference
9
6 Application Examples
In order to highlight and illustrate the topology optimization features of MSC/NASTRAN-OPTISHAPE,
several examples are provided here. Although not all capabilities are demonstrated with these
examples, these examples cover some important features that are used.
6.1 A 2D Plate under 3 Concentrated Forces
The first example is a very simple 2D plate under 3 concentrated loading cases as shown in Fig. 5. The
constraint volume is 0.3, which means 70% of the material in the design domain is going to be removed.
The design domain is modeled with 800 CQUAD4 elements, and the 3 concentrated loading forces are
applied to the model with 3 subcases. Table 2 shows part of input bulk data deck. By looking at the
bulk data deck, one can see that there are 3 changes/modifications besides the common static analysis
data as follows:
1) assign userfile='s2dl3.inp',unit=31,form=formatted,status=unknown to assign a file for element volume density ratios output; 2) sol topopt to select topology optimization sequence; 3) dti topol 0 1 2 1 30 .3 0.5 .001 5
dti topol 1 1 1 endrec to provide the topology optimization parameters, define the design domain. etc..
Figure 5 A 2D Plate under 3 Concentrated Loading Forces
Case1 Case2 Case3
L=400.0
W=200.0E=2.90E+7v=0.32
t=1.0
X
Y
Nodal Force
Design Domain
10
Table 2 Input Bulk Data
ID TEST2D, S2D800 assign userfile='s2dl3.inp',unit=31,form=formatted,status=unknown sol topopt CEND … … SPC = 1 SUBCASE 1 LOAD = 1 SUBCASE 2 LOAD = 2 SUBCASE 3 LOAD = 3 $ BEGIN BULK PARAM POST -1 PARAM AUTOSPC YES $ADDITIONAL BULK DATA FOR STATIC TOPOLOGY OPTIMIZATION dti topol 0 1 2 1 30 .3 0.5 .001 5 dti topol 1 1 1 endrec $ PSHELL 1 1 1. 1 1 CQUAD4 1 1 1 2 43 42 … … … … all other data to define element connection, grid points, boundary and loading forces etc. … … FORCE 3 31 0 1. 0. -1. 0. ENDDATA
Figure 6 shows the resultant topology obtained from MSC/NASTRAN-OPTISHAPE. In this figure, these
elements with element volume density ratios smaller than 0.3 are hidden.
Figure 6 Resultant Topology of a 2D Plate under 3 Loading Cases
11
6.2 Partial Domain Design of an I Beam Structure The second example is used to illustrate the features when 1) only partial structure design is necessary;
2) the design domain can be composed of “composite” shell with a middle base ply which remains
unchanged, which is usually used in the design of a shell structure with distributing reinforcement.
The volume constraint is 0.3. Figure 7 shows the model, loading and boundary conditions, and final
resultant topology.
Figure 7 Partial Domain Design of an I Beam Structure
6.3 A 3D Universal Joint Design The third example is a 3D static topology optimization example for a universal joint. Only a half model
is used here due to symmetry condition. In this analysis, 9932 CHEXA and 1 RBE2 elements are used.
Figures 8-10 show the analysis model, the design domain and the resultant topology, respectively.
Figure 8 Analysis Model
Rigid Element
Force1
Load case1
Rigid Element
Force1
Force2
Load case2
Design Domain
No Design
No Design
H=5.0
W=4.0
t
t0
L=20.0
E=21000.0v =0.3
t =1.0t0=0.2
No Design
No DesignNodal force
12
Figure 9 Finite Element Mesh and Design Domain
Figure 10 Resultant Topology of the Universal Joint
6.4 Normal Modes Topology Optimization of a Copy Machine Stand
The forth example is a normal mode topology optimization applied to a copy machine stand in order to
find the optimal distributing reinforcement. In this analysis, 1 CONM2, 1 RBE2 and 6852 CQUAD4
elements are used with 2060 CQUAD4 element being designed. The objective function is to maximize
the first 6 eigenvalues subject to a volume constraint of 50%, which means 50% of the material in the
design domain is to be removed. Figure 11 shows the analysis model. Figure 12 shows both the resultant
topology of the copy machine stand and its changing history of the first 6 eigenvalues. Please note that
the eigenvalues obtained after the first design cycle are the values of the structure with 50% of the
material removed uniformly from all design elements.
Design domain
Non-design domain
13
Figure 11 Normal Modes Topology Optimization Model of a Copy Machine Stand
Figure 12 Resultant Topology and History of First 6 Eigenvalues
7 Concluding Remarks
In this development, by combining the topology optimization optimizer function of OPTISHAPE into
MSC/NASTRAN as functional modules, both static and normal modes topology optimization analysis
capabilities are introduced into MSC/NASTRAN as a special solution sequence. The principal features
of the new capabilities are as follows:
0 10 20 3010
20
30
14
a) Static topology optimization : To minimize mean compliance subject to volume constraint.
b) Normal modes topology optimization : To maximize eigenfrequencies in appointed order or
minimize the difference between given eigenfrequencies and the calculated ones subject to volume
constraint.
c) Shell or solid design elements : Either first-order shell or solid elements can be used as design
elements. All other elements that can be used in linear static or normal modes analysis can also be
used to model the non-design part of the optimization analysis model
d)MSC/PATRAN integration : A special MSC/PATRAN preference, MSC/NASTRAN–OPTISHAPE
preference, has been developed. With this preference, all pre and post processing for both static
and normal modes topology optimization analyses can be carried out within MSC/PATRAN.
e) MSC/NASTRAN bulk data format: All input bulk data are provided in MSC/NASTRAN bulk
data format – compared with the common MSC/NASTRAN static or normal modes analysis jobs,
only a few additional data lines are sufficient to perform the associated static or normal modes
topology analysis. These additional data lines can be added by MSC/PATRAN. It is also possible
and quite easy to manually insert these additional lines once the common MSC/NASTRAN static
or normal modes bulk data are generated by any other preprocessors.
f) Efficient solution of very large models: By using MSC/NASTRAN advanced elements and
efficient solver engines for static or normal modes analysis, very efficient solution to very large
scale topology optimization models can be realized.
Acknowledgements
The authors would like to thank Mr. Gopal K. Nagendra of MSC India, Mr. Qiwen Liu of DongFeng
Motor Corporation, and Mr. JiDong Yang of MSC Japan, Ltd. for their great help in the development of
MSC/NASTRAN-OPTISHAPE.
References
[1] Bendsoe, M.P.,Kikuchi,N. : Comput. Methods Appl. Mech. Eng., 71 (1988), 197.
[2] Suzuki,K., Kikuchi,N. : Comput. Methods Appl. Mech. Eng., 93 (1991), 291.
[3] Diaz, A.R., Kikuchi,N. : Int. J. Numer. Methods Eng., 35 (1992), 1487.